Finite Group Databasebeta

← #22602452

Subgroups — #22602452

Order 9088.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986734C213yesmaximal, complemented
2#17986715C213yesmaximal, complemented
3#17986702C213yesmaximal, complemented
4#17986687C213yesmaximal, complemented
5#17986671C211yesmaximal, characteristic, complemented
6#17986661C211yesmaximal, characteristic, complemented
7#17986661C211yesmaximal, characteristic, complemented
8#17986623C213yesmaximal, complemented
9#17986605C213yesmaximal, complemented
10#17986536C213yesmaximal, complemented
11#17986534C213yesmaximal, complemented
12#17986481C211yesmaximal, characteristic, complemented
13#17986475C211yesmaximal, characteristic, complemented
14C4×#11765866C211yesmaximal, characteristic, complemented
15#557×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C2×#11765866C2211yescharacteristic
18C2×Q8×D71C2211yescharacteristic
19#14556125C2213yes—
20#14556123C2216yes—
21#14556123C2213yes—
22#14556123C2211yescharacteristic
23#14556119C2213yes—
24#14556117C2216yes—
25#14556117C2213yes—
26#14556116C2213yes—
27#14556116C2213yes—
28#14556113C2216yes—
29#14556113C2216yes—
30#14556113C2213yes—
31#14556113C2213yes—
32#14556108C2213yes—
33#14556108C2213yes—
34#14556104C2213yes—
35#14556103C2213yes—
36#14556103C2213yes—
37#14556103C2213yes—
38#14556103C2216yes—
39#14556103C2213yes—
40#14556103C2213yes—
41Q8×Dic71C2211yescharacteristic
42C4×Dic142C2213yes—
43#14556093C2213yes—
44C4×D284C2213yes—
45C4×D284C2213yes—
46C4×D284C2213yes—
47C4×D284C2213yes—
48C4×D284C2213yes—
49#14556090C2213yes—
50#14556090C2213yes—
51#14556090C2216yes—
52#14556086C2213yes—
53#14556086C2216yes—
54#14556086C2213yes—
55#14556085C2213yes—
56#14556085C2213yes—
57C4⋊C4×D71C2213yes—
58C42×D71C2213yes—
59C4⋊Q8×C71C2213yes—
60#125×C71C2213yes—
61#125×C71C2213yes—
62C4×Q8×C71C2211yescharacteristic
63C4×Q8×C71C2211yescharacteristic
64C4×Q8×C71C2211yescharacteristic
65C4×Q8×C71C2213yes—
66#11765866—24no—
67#11765866—24no—
68C2×Dic142C2313yes—
69Q8×D71—24no—
70C2×D284C2313yes—
71C2×D284C2313yes—
72C2×D284C2316yes—
73C2×D284C2313yes—
74C2×D284C2313yes—
75C2×D284C2313yes—
76C2×C4×D71C2313yes—
77C2×C4×D71C2313yes—
78C2×C4×D71C2313yes—
79C2×C4×D71C2313yes—
80C2×C4×D71C2313yes—
81C2×C4×D71C2316yes—
82C4⋊Dic71C2313yes—
83C4⋊Dic71C2313yes—
84C4⋊Dic71C2311yescharacteristic
85C4⋊Dic71C2313yes—
86Dic71⋊C4C2313yes—
87Dic71⋊C4C2311yescharacteristic
88Dic71⋊C4C2313yes—
89Dic71⋊C4C2313yes—
90(C2×D71)⋊C4C2311yescharacteristic
91(C2×D71)⋊C4C2313yes—
92(C2×D71)⋊C4C2313yes—
93(C2×D71)⋊C4C2313yes—
94(C2×D71)⋊C4C2313yes—
95(C2×D71)⋊C4C2316yes—
96(C2×D71)⋊C4C2316yes—
97(C2×D71)⋊C4C2313yes—
98(C2×D71)⋊C4C2316yes—
99(C2×D71)⋊C4C2311yescharacteristic
100(C2×D71)⋊C4C2311yescharacteristic
101(C2×D71)⋊C4C2313yes—
102(C2×D71)⋊C4C2313yes—
103C4×Dic71C2311yescharacteristic
104C4×Dic71C2313yes—
105C4×Dic71C2313yes—
106C4×Dic71C2316yes—
107C2×Q8×C71C2313yes—
108C2×Q8×C71C2311yescharacteristic
109C4⋊C4×C71C2311yescharacteristic
110C4⋊C4×C71C2313yes—
111C4⋊C4×C71C2313yes—
112C4⋊C4×C71C2313yes—
113C4⋊C4×C71C2313yes—
114C4⋊C4×C71C2313yes—
115C4⋊C4×C71C2316yes—
116C42×C71C2313yes—
117C42×C71C2313yes—
118C42×C71C2313yes—
119C22×D71C2411yescharacteristic
120C22×D71C2413yes—
121C22×D71C2413yes—
122Q8×C71—43no—
123Q8×C71#4114yes—
124C2×C4×C71C2411yescharacteristic
125C2×C4×C71C2411yescharacteristic
126C2×C4×C71C2411yescharacteristic
127C2×C4×C71C2413yes—
128C2×C4×C71C2413yes—
129C2×C4×C71C2413yes—
130C2×C4×C71C2413yes—
131C2×Dic71C2411yescharacteristic
132C2×Dic71C2413yes—
133C2×Dic71C2411yescharacteristic
134C2×Dic71C2413yes—
135C2×Dic71C2411yescharacteristic
136D284—26no—
137D284—26no—
138D284—43no—
139D284—43no—
140D284—43no—
141D284—46no—
142C4×D71—26no—
143C4×D71—26no—
144C4×D71—26no—
145C4×D71—43no—
146C4×D71—46no—
147C4×D71—43no—
148Dic142—26no—
149C22×C71C2511yescharacteristic
150C2×D71—21no—
151C2×D71—23no—
152C2×D71—23no—
153C2×D71—23no—
154C2×D71—23no—
155C2×D71—23no—
156C2×D71—21no—
157C2×D71—23no—
158C2×D71—21no—
159C4×C71—21no—
160C4×C71—23no—
161C4×C71—23no—
162C4×C71—21no—
163C4×C71—23no—
164C4×C71—21no—
165C4×C71C2×#4116yes—
166Dic71—23no—
167Dic71—21no—
168Dic71—21no—
169Dic71—21no—
170Dic71—23no—
171C2×C71C22×#4111yescharacteristic
172C2×C71#58511yescharacteristic
173C2×C71#58511yescharacteristic
174D71—41no—
175D71—43no—
176D71—43no—
177#3524—711nomaximal, Sylow
178C71#352411yescharacteristic, complemented, Sylow
179#559—713no—
180#557D7111yescharacteristic, complemented
181#555—713no—
182#553—713no—
183#552—713no—
184#550—711no—
185#548—711no—
186#548—711no—
187#542—713no—
188#540—713no—
189#529—713no—
190#529—713no—
191#520—711no—
192#519—711no—
193C4×#41—711no—
194C2×#41—711no—
195C2×#41—711no—
196C22×Q8—711no—
197C4⋊Q8C2×D7113yes—
198#127—713no—
199#126—711no—
200#126—713no—
201#126—716no—
202#125—713no—
203#125C2×D7113yes—
204#125C2×D7113yes—
205#124—713no—
206#124—716no—
207#124—713no—
208#124—713no—
209#123—716no—
210#123—713no—
211#123—713no—
212#123—716no—
213C22⋊Q8—713no—
214C22⋊Q8—713no—
215C22⋊Q8—713no—
216#121—713no—
217#121—713no—
218#121—713no—
219#121—713no—
220#121—716no—
221#121—713no—
222C4×Q8—711no—
223C4×Q8—713no—
224C4×Q8—713no—
225C4×Q8C2×D7111yescharacteristic
226C4×Q8C2×D7111yescharacteristic
227C4×Q8C2×D7111yescharacteristic
228C4×Q8C2×D7113yes—
229C4×D4—713no—
230C4×D4—713no—
231C4×D4—716no—
232C4×D4—713no—
233C4×D4—713no—
234C4×D4—713no—
235C4×D4—713no—
236C4×D4—713no—
237#117—713no—
238#117—713no—
239#117—716no—
240#117—713no—
241#117—713no—
242C2×C4⋊C4—713no—
243C2×C42—713no—
244#41—1424no—
245#41—1424no—
246C2×Q8—713no—
247C2×Q8—1424no—
248C2×Q8C22×D7113yes—
249C2×Q8C22×D7111yescharacteristic
250C2×D4—713no—
251C2×D4—713no—
252C2×D4—716no—
253C2×D4—713no—
254C2×D4—713no—
255C2×D4—713no—
256C22×C4—713no—
257C22×C4—713no—
258C22×C4—716no—
259C22×C4—713no—
260C22×C4—713no—
261C22×C4—713no—
262C4⋊C4—713no—
263C4⋊C4—711no—
264C4⋊C4—713no—
265C4⋊C4—713no—
266C4⋊C4—713no—
267C4⋊C4—713no—
268C4⋊C4—711no—
269C4⋊C4—713no—
270C4⋊C4C22×D7113yes—
271C4⋊C4C22×D7113yes—
272C4⋊C4C22×D7113yes—
273C4⋊C4C22×D7116yes—
274C4⋊C4C22×D7113yes—
275C4⋊C4C22×D7111yescharacteristic
276C4⋊C4C22×D7113yes—
277C22⋊C4—711no—
278C22⋊C4—713no—
279C22⋊C4—713no—
280C22⋊C4—713no—
281C22⋊C4—716no—
282C22⋊C4—713no—
283C22⋊C4—716no—
284C22⋊C4—711no—
285C22⋊C4—713no—
286C22⋊C4—711no—
287C22⋊C4—713no—
288C22⋊C4—716no—
289C22⋊C4—713no—
290C42—711no—
291C42—713no—
292C42—713no—
293C42—716no—
294C42C22×D7113yes—
295C42C22×D7113yes—
296C42C22×D7113yes—
297C23—711no—
298C23—713no—
299C23—713no—
300Q8—43no—
301Q8—1426no—
302Q8#1176586614yes—
303D4—2843no—
304D4—2846no—
305D4—2843no—
306D4—2843no—
307D4—1426no—
308D4—1426no—
309C2×C4—2843no—
310C2×C4—2843no—
311C2×C4—2846no—
312C2×C4—711no—
313C2×C4—711no—
314C2×C4—713no—
315C2×C4—713no—
316C2×C4—711no—
317C2×C4—1426no—
318C2×C4—1426no—
319C2×C4—1426no—
320C2×C4C23×D7113yes—
321C2×C4C23×D7111yescharacteristic
322C2×C4C23×D7113yes—
323C2×C4C23×D7111yescharacteristic
324C2×C4C23×D7111yescharacteristic
325C2×C4C23×D7113yes—
326C2×C4C23×D7113yes—
327C22—1421no—
328C22—1423no—
329C22—1423no—
330C22—1423no—
331C22—1421no—
332C22—1423no—
333C22—1423no—
334C22—1423no—
335C22—1421no—
336C22C24×D7111yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—1421no—
344C4—1421no—
345C4—1423no—
346C4—1421no—
347C4—1423no—
348C4C2×#1176586616yes—
349C2—2841no—
350C2—2843no—
351C2—2843no—
352C2C22×#1176586611yescharacteristic
353C2#1798683511yescharacteristic
354C2#1798683611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D71
commutator, socleC22×C71C25
Fitting#557×C71C2
hypercenter#557D71
hypocenterC71#3524
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ #557

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #17986734C2semidirect
2. #17986715C2semidirect
3. #17986702C2semidirect
4. #17986687C2semidirect
5. #17986671C2semidirect
6. #17986661C2semidirect
7. #17986661C2semidirect
8. #17986623C2semidirect
9. #17986605C2semidirect
10. #17986536C2semidirect
11. #17986534C2semidirect
12. #17986481C2semidirect
13. #17986475C2semidirect
14. C4×#11765866C2semidirect
15. #557×C71C2semidirect
178. C71#3524semidirect
180. #557D71semidirect